Abstract

This paper discusses oscillatory and asymptotic behavior of solutions of a class
of third-order nonlinear functional differential equations. By using the generalized Riccati
transformation and the integral averaging technique, three new sufficient conditions which
insure that the solution is oscillatory or converges to zero are established. The results obtained
essentially generalize and improve the earlier ones.

1. Introduction

As is well known, the comparison and separation theory of zeros distribution for second-order homogeneous linear differential equations established by Ladde et al. lays a foundation of oscillation theory for differential equations. During one and a half century, oscillation theory of differential equations has developed quickly and played an important role in qualitative theory of differential equations and the theory of boundary value problem. Oscillation theory of differential equations has been widely used in areas of physics, mechanics, radio technology, control system, sciences of life, economic relations, and population growth. The oscillations are physical phenomena which widely exist in physics and technological sciences, such as the oscillation of building and machine, electromagnetic vibration in radio technology and optical science, self-excited vibration in control system, sound vibration, beam vibration in synchrotron accelerator, the vibration sparked for burning rocket engine, and the complicated oscillation in chemical reaction. All different phenomena can be unified into an oscillation theory through an oscillation equation. There are many books on the oscillation theory, about which we can refer to [1].

The oscillation theory of third-order nonlinear functional differential equations has been widely applied in research of a lossless high-speed computer network and physical sciences. In this paper, we are concerned with oscillatory behavior of a third-order nonlinear functional differential equation as follows:
where is the ratio of positive odd integers. We have the following hypotheses: (A1) and satisfy
(A2) , such that , ; (A3) , , and .

By a solution of (1), we mean a nontrivial function satisfying (1) which has the properties for and . Our attention is paid to those solutions of (1) which satisfy for all . A solution of (1) is said to be oscillatory on if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

In recent years, there have been numerous researches or many research activities concerning the oscillation and nonoscillation of solutions of three-order functional differential equations, which are special cases of (1), and for recent contributions, we refer to [2–8]. Consider

Parhi and Padhi [2] studied asymptotic behavior of solutions of (3). By using the integral averaging technique, Baculíková et al. [3] obtained sufficient conditions which insured that the solution of self-liner ordinary differential equation (4) was oscillatory or converges to zero. Mojsej [4] established the comparison results which insured that the solution of (5) was oscillatory or converges to zero. By the integral averaging technique, Saker [5] gave some oscillatory results of (5) when the condition holds. Several authors had proved some oscillatory results of (6) by method of comparison; see [6–8]. In this paper we intend to use Riccati transformation and the integral averaging technique to obtain some sufficient conditions which guarantee that every solution of (1) is oscillatory or converges to zero. Our results generalize and improve the corresponding theorems established in [3, 5].

2. Several Lemmas

Lemma 1. Assume that is a positive solution of (1). Then, there exists such that either (I) , , , ,or (II) , , , .

Lemma 2. Assume that is a positive solution of (1) which satisfies case (I) in Lemma 1. Then there exists , such that
Assume that (8) and hold. Then

Proof. Pick so that for . Using (1), we obtain
Then, is strictly decreasing on . We get
and, hence, we have
By integrating both sides of the above inequality from to , it yields
Furthermore, by integrating both sides of (1) from to and noting that , , , we obtain
Then,
This completes the proof.

Lemma 3. Assume that is a positive solution of (1) which satisfies case (II) in Lemma 1. Furthermore,
Then, .

Proof. Assume that is a positive solution of (1) which satisfies case (II) in Lemma 1. Then, is decreasing and . We assert that . If not, then , . Integrating (1) from to , we get
Hence, we have
Integrating the above inequality from to , we obtain
Integrating the last inequality again from to , we have
Since condition (18) holds, we obtain , which contradicts . Hence, . This completes the proof.

3. Main Results

In this section, we obtain three new oscillatory criteria for (1) by using the generalized Riccati transformation and integral averaging technique of Philos-type [9]. Let

A function is said to have the property of and denote if it satisfies(i), ; , ;(ii) and it is continuous.

The following are the main results of this paper.

Theorem 4. Let (8), (18), and hold. Assume that there exist , , and , such that
and for arbitrary , one has
where
Then, every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists a sufficiently large , such that , . By Lemma 1, we see that satisfies either case (I) or case (II).If case (I) holds, then , . Define the function by
Then,
When holds, using (9) and (11), we get
In view of (28) and (29), noting that , we obtain
When holds, using (9) and (10), we have
In view of (28) and (31), which yields
From (30), (32), and the definition of , we get
By the definition of , we have
From (33) and (34), noting the definition of and , we obtain
Multiplying both sides of (35), with replaced by , by , integrating with respect to from to , we get
By integrating parts and using and (24), we obtain
Using averaging technique, we have
Combining (37) and (38), we get
which contradicts (25).If case (II) holds, from (18), by Lemma 3, . This completes the proof.

Theorem 5. Let (8), (18), and hold. Assuming that there exist and, for all sufficiently large , there exists a , one has
where , , and are defined in Theorem 4. Then every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that there exists a sufficiently large , such that , . By Lemma 1, we see that satisfies either case (I) or case (II).If case (I) holds, we proceed in the proof of Theorem 4 and get (34). Then, from the definition of and , we obtain
By using the averaging technique, we find that
Hence, we get
Integrating (43) from to , we have
It follows that
which contradicts (40).If case (II) holds, from (18), by Lemma 3, . This completes the proof.

Corollary 6. Let (8), (18), and hold, and for all sufficiently large , there exists a ; then, one has
where is defined in Theorem 4. Then every solution of (1) is oscillatory or converges to zero.

Theorem 7. Let (18) and hold. Assume that there exist , P, and , such that
and all sufficiently large such that
Then, every solution of (1) is oscillatory or converges to zero.

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that is a positive solution of (1). By Lemma 1, we see that satisfies either case (I) or case (II).If case (I) holds, then , . Define the function by
Using (9), we have
Hence, by the definition of , we have
Multiplying both sides of (51), with replaced by , by and integrating with respect to from to , we get
Integrating by parts and using (47), which yields
Define and as follows:
where , , and . Using the inequality [10, Theorem 41]
we obtain
Combining (53) and (56), we get
which contradicts (48).If case (II) holds, from (18), by Lemma 3, . This completes the proof.

Remark 8. If we let , in Theorem 4 and the function is of Theorem 3.3 in [5], then condition (25) is in [5]. Therefore, the result of in [5] is generalized to the case that is the ratio of positive odd integers. If we let in Theorem 7, the function is of Theorem 3.4 in [3], which condition (48) is converted to in [3]. Then, the result of in [3] is generalized to the one of (1) in this paper.

Example 9. Consider the three-order differential equation
where
Conditions (A1), (A2), and (A3) are clearly satisfied. It is easy to find that (8) and (18) hold. Let , . Here
From Theorem 5, we have
so (40) is satisfied. Hence, by Theorem 5, every solution of (58) is oscillatory or converges to zero.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2013AM003), the Development Program in Science and Technology of Shandong Province of China (2010GWZ20401), and the Science Foundation of Binzhou University (BZXYKJ0810).